Other Single-Loop Circuits — Question 5

Question 5

A monotone nonlinear load gives one physical operating current

An ideal 14.0 V source drives a 1.00 ohm resistor and a nonlinear load with $V_d(I)=(2.00\,\Omega)I+(1.00\,\mathrm{V/A^3})I^3$.

1. Find the steady current and prove uniqueness.

2. Find the small current change for a small emf change deltaE near that state.

Question 5 - Solution

A monotone nonlinear load gives one physical operating current

(a) Method and calculation

$$14.0=I(1.00)+V_d(I)\Rightarrow I^3+3I-14=0\quad(I\text{ in A}).$$$$I=2.00\,\mathrm A,\quad d(I^3+3I)/dI=3I^2+3>0.$$

(b) Interpretation and further calculation

$$\delta E=[1.00+2.00+3(1.00)I^2]\,\delta I.$$$$I=2.00\,\mathrm A:\quad\delta I\simeq\delta E/(15.0\,\Omega).$$

Physical check

A strictly increasing total voltage requirement intersects a fixed source voltage at exactly one current.